What are the divisors of 3193?

1, 31, 103, 3193

There is a total of 4 positive divisors.
The sum of these divisors is 3328.
The arithmetic mean is 832.

4 odd divisors

1, 31, 103, 3193

How to compute the divisors of 3193?

A number N is said to be divisible by a number M (with M non-zero) if, when we divide N by M, the remainder of the division is zero.

$N mod M = 0$

Brute force algorithm

We could start by using a brute-force method which would involve dividing 3193 by each of the numbers from 1 to 3193 to determine which ones have a remainder equal to 0.

$Remainder = N - (M \times ⌊ \frac{N}{M} ⌋)$

(where $⌊ \frac{N}{M} ⌋$ is the integer part of the quotient)

3193 / 1 = 3193 (the remainder is 0, so 1 is a divisor of 3193)
3193 / 2 = 1596.5 (the remainder is 1, so 2 is not a divisor of 3193)
3193 / 3 = 1064.3333333333 (the remainder is 1, so 3 is not a divisor of 3193)
...
3193 / 3192 = 1.000313283208 (the remainder is 1, so 3192 is not a divisor of 3193)
3193 / 3193 = 1 (the remainder is 0, so 3193 is a divisor of 3193)

Improved algorithm using square-root

However, there is another slightly better approach that reduces the number of iterations by testing only integers less than or equal to the square root of 3193 (i.e. 56.506636778347). Indeed, if a number N has a divisor D greater than its square root, then there is necessarily a smaller divisor d such that:

$D \times d = N$

(thus, if $\frac{N}{D} = d$ , then $\frac{N}{d} = D$ )

3193 / 1 = 3193 (the remainder is 0, so 1 and 3193 are divisors of 3193)
3193 / 2 = 1596.5 (the remainder is 1, so 2 is not a divisor of 3193)
3193 / 3 = 1064.3333333333 (the remainder is 1, so 3 is not a divisor of 3193)
...
3193 / 55 = 58.054545454545 (the remainder is 3, so 55 is not a divisor of 3193)
3193 / 56 = 57.017857142857 (the remainder is 1, so 56 is not a divisor of 3193)